Formation of proto-clusters and star formation within clusters: apparent universality of the initial mass function ?

It is believed that the majority of stars form in clusters. Therefore it is likely that the gas physical conditions that prevail in forming clusters, largely determine the properties of stars that form and in particular the initial mass function. We develop an analytical model to account for the formation of low mass clusters and the formation of stars within clusters. The formation of clusters is determined by an accretion rate, the virial equilibrium and energy as well as thermal balance. For this latter both molecular and dust cooling are considered using published rates. The star distribution is computed within the cluster using the physical conditions inferred from this model and the Hennebelle & Chabrier theory. Our model reproduces well the mass-size relation of low mass clusters (up to few $\simeq 10^3$ M$_\odot$ of stars corresponding to about 5 times more gas) and an initial mass function which is $i)$ very close to the Chabrier's IMF, $ii)$ weakly dependent on the mass of the clusters, $iii)$ relatively robust to (i.e. not too steeply dependent on) variations of physical quantities as accretion rate, radiation and cosmic rays abundances. The weak dependence of the mass distribution of stars with the cluster mass results from the compensation between varying clusters densities, velocity dispersions and temperatures all inferred from first physical principles. This constitutes a possible explanation for the apparent universality of the IMF within the Galaxy though variations with the local conditions could certainly be observed.